Invertible elements of alternative unital ring form Moufang loop

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Statement

Suppose R is an alternative unital ring with multiplication \! *. Suppose S is the subset of R comprising those elements of R that possess two-sided inverses for \! *. Then, S is closed under * and acquires the structure of a Moufang loop (?) under *.

Related facts

Facts used

  1. Alternative ring satisfies Moufang identities

Proof

The proof follows directly from fact (1).